New slope inequalities for families of complete intersections

TL;DR

This paper establishes new slope inequalities for families of complete intersections, providing criteria for positivity and stability conditions in algebraic geometry.

Contribution

It proves $f$-positivity of $ ext{O}_X(1)$ for fibrations with complete intersection fibers and derives numerical conditions for positivity and stability.

Findings

Proves $f$-positivity of $ ext{O}_X(1)$ for arbitrary dimension fibrations.

Provides necessary and sufficient conditions for positivity of powers of $ ext{O}_X(1)$ and the relative canonical sheaf.

Derives Chow instability conditions for fibers in projective bundles of $ ext{mu}$-unstable bundles.

Abstract

We prove $f$-positivity of $\mathcal{O}_X(1)$ for arbitrary dimension fibrations over curves $f\colon X\to B$ whose general fibre is a complete intersection. In the special case where the family is a global complete intersection, we prove numerical sufficient and necessary conditions for $f$-positivity of powers of $\mathcal{O}_X(1)$ and for the relative canonical sheaf. From these results we also derive a Chow instability condition for the fibres of relative complete intersections in the projective bundle of a $\mu-$unstable bundle.